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B0j mathcad/worksheet.xml ( Converted from "Inventory Control Example in Mathcad" created by Piet WattéThe news vendor problemP. Watté, piet.watte@skynet.be1. IntroductionThe following problem is a classical example of inventory control in operational research. A news magazine vendor wants to determine the number of copies of a journal he has to source from a publisher each week. He sells the journals on Sunday, without knowing the exact demand, whereby he has to place weekly a fixed order (same amount of journals throughout the year). Historical data of the past year sales show that the number of journals he sold was normally distributed with a mean equal to 12 and a standard deviation of 3. The associated histogram is depicted in figure 1.He purchases each copy of the journal for 25 cents, which he can sell for 75 cents (i.e. with a contribution margin of 50 cents). Unsold copies can be returned to the publisher for 10 cents.This is a typical example of an inventory control problem with uncertain demand. A single product is to be ordered at the beginning of a period and can only be used to satisfy demand during that period.Define c0 as the cost per unit of positive inventory, or the overage cost. The overage cost is here is this case equal to -(25-10) or -15 centsIdem for cu, as the cost per unit of unsatisfied demand (=cost of a negative ending inventory), known as the underage cost. Here the overage cost is 75-25 or 50 centsHereby we assume that the demand D is a continuous non-negative variable, which has a density function F(x), here taken as a Gaussian distribution function.Define following function:with Q the size of the order the newsboy places every week and D the demand variable which is normally distributedThe benefits can be calculated as:It can be shown [1-2] that the expected value of the costs is equal toEq. 1Using some calculus, we find the optimum order quantity Qopt as the value for which the costs G(Q) are minimum. This is realised when the derivative dG/dQ is equal to zero orEq 2Or the optimum order quantity can be found as Qopt for whichEq 3or withthe inverse cumulative standard normal distribution function. 2. Analytical calculationIn this example, the overage costs, or the costs of a positive inventory are equal to:The underage costs, are the costs of a unsatisfied demand and are equal to:From historical data, we know that the number of newspapers is distributed over the 52 weeks of a year with a mean µ and standard deviation sThus, the optimum order quantity is equal to 14. Or in other words, if the news vendor orders a fixed amount of 14 copies for each week-end, he maximizes its profits.3. Numerical calculationThe above calculation, can also be performed numerically.Let us now determine the optimum order point, using random number generation and compare the outcome, with the analytical resultDrawing random numbers form a normal distribution with mean µ and standard deviation s can be done in Mathcad using the function cnorm(µ,s).Following command generates a vector of such valuesfigure 1: histogram of the number of journals sold each weekcolumns12012Hereafter we define the function benefits. It calculates the bottom line for an ordering point of size equal to 'size', in account of the earlier defined under and overage costs.Note that in the function above, we round the random numbers to integers. The third line, is the numerical equivalent of the integral in the formula Eq. 1Figure 2: cost function of the news vendor, in account of the over and underage costs.lines1103655125From this graph it is readily that the optimum lot size of journals that has to be purchased each week should be equal to 14 (where the cost is minimum). Thus, there is a good agreement between the analytical and the numerical approach.4. Exercise Examine the optimum order point for variations in the overage and underage costs and combine this with different values of µ and s for the demand. Verify whether the numerical and the analytical outcome still match. Investigate the influence of the number of iterations in the Monte Carlo simulation (here totalnumber)Instead of using the mathcad function cnorm, random numbers from a normal distribution can also be drawn using [2]:
with U1 and U2, two draws from (0,1) uniform distributionVerify whether you still obtain the same results5. References[1] S. Nahmias, "Production and operation analysis", IRWIN, (1993), p 244[2] W J. Hopp and M. Spearman, "Factory Physics", Mc Graw Hill, (1999) , p65PK
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